Hilbert-Glass Transition: New Universality of Temperature-Tuned Many-Body Dynamical Quantum Criticality

We study a new class of unconventional critical phenomena that is characterized by singularities only in dynamical quantities and has no thermodynamic signatures. One example of such a transition is the recently proposed many-body localization-delocalization transition, in which transport coefficients vanish at a critical temperature with no singularities in thermodynamic observables. Describing this purely dynamical quantum criticality is technically challenging as understanding the finite-temperature dynamics necessarily requires averaging over a large number of matrix elements between many-body eigenstates. Here, we develop a real-space renormalization group method for excited states that allows us to overcome this challenge in a large class of models. We characterize a specific example: the 1 D disordered transverse-field Ising model with generic interactions. While thermodynamic phase transitions are generally forbidden in this model, using the real-space renormalization group method for excited states we find a finite-temperature dynamical transition between two localized phases. The transition is characterized by nonanalyticities in the low-frequency heat conductivity and in the long-time (dynamic) spin correlation function. The latter is a consequence of an up-down spin symmetry that results in the appearance of an Edwards-Anderson-like order parameter in one of the localized phases.

Popular Summary

Upon heating, a compass needle loses its magnetization, or a liquid turns into vapor, at a particular material-dependent temperature. These are the two everyday examples of conventional phase transitions. These known phase transitions all have one property in common: There is at least one thermodynamic observable undergoing a qualitative or dramatic change, such as the magnetization in the case of a compass needle or the density in the case of a liquid. In this theoretical paper, we demonstrate a new type of quantum phase transition that leaves no signature in any thermodynamic observable, showing a fundamental departure from the conventional types.

The system where we find such an unusual phase transition is actually rather simple: a one-dimensional chain of quantum spins with nearest-neighbor interactions and local magnetic fields, both of random magnitude. The disorder immobilizes magnetic excitations, and this effect ultimately enables a novel athermal symmetry-breaking phase transition. An analysis of this phase transition is challenging since the transition involves nonequilibrium quantum properties of the system rather than its ground state, and its signatures through time-dependent transport properties, such as low-frequency heat conductivity, become unambiguously clear only in large systems..

Nonequilibrium behavior of a quantum system is determined by its excitations; the key is therefore to gain the knowledge of the excited many-spin eigenstates of a large system with disorder. Up to now, only systems with fewer than 100 spins could be reliably explored. Using the real-space renormalization-group method that we have developed for this purpose, however, we are able to construct excited eigenstates of systems having thousands of spins and hence obtain definitive results.

The transition we have analyzed is perhaps the simplest member of a new class of quantum dynamical transitions that could be realized in ultracold atom systems or, utilizing ultrafast lasers, in condensed matter systems. Our work opens the door to future exploration and classification.

Figure 1

(a) Schematic representation of a typical eigenstate in the paramagnetic (local-field-dominated) phase: the local spins are largely aligned (i) or antialigned (ii) with the local fields; rare clusters can still be present (iii). (b) Hilbert-glass (cluster-dominated) eigenstate: local spins form magnetic clusters that often contain domain walls; rare isolated spins can still be present. (c) Schematic phase diagram of the Hilbert-glass transition (HGT) in the tuning parameter–temperature plane. The diagram shows the HGT line separating the paramagnetic (local-field-dominated) and the Hilbert-glass (cluster-dominated) phases terminating in a zero-temperature quantum critical point (QCP). The behavior of equilibrium thermodynamic observables at finite temperatures is governed by the QCP in the region of critical fluctuations [15]. While equilibrium thermodynamic observables have no singularities in the phase diagram, with the exception of the zero-temperature QCP, dynamical observables show singular behavior along the HGT line.

Figure 2

(a) The RSRG-X tree for a six-site Ising chain. The leaves of the tree correspond to the many-body eigenstates of the spin chain. (b) The corresponding eigenspectrum found by exact diagonalization. (c),(d) RSRG-X rules for site (c) and bond (d) decimation in the ${\mathrm{hJJ}}^{\prime }$ model. In the site decimation rule, a local magnetic field ($h_2$) is the dominant coupling; thus, the corresponding local spin is either aligned (ground state) or antialigned (excited state) with it. Eliminating $h_2$, we obtain a new set of couplings ${\tilde{h}}_1$, ${\tilde{J}}_1$, ${{\tilde{J}}_1}^{\prime }$, and ${\tilde{h}}_3$ to second order (see Appendix pp2 for details). In the bond decimation rule, $J_2$ is the dominant coupling, corresponding to the neighboring spins being either aligned (ground state) or antialigned (excited state, i.e., a domain wall) along the $x$ axis.

Figure 3

(a) Renormalization group flows in the $\mathcal{C}\text{−}\mathcal{D}$ plane, where $\mathcal{C}$ measures whether local fields or bonds are stronger while $\mathcal{D}$ measures the strength of disorder. Tuning $\beta$ results in RSRG-X flows towards the Hilbert-glass phase, the quantum critical point, and the paramagnetic phase (PM), as indicated by arrows. (The initial value of the tuning parameter $\delta$ and the width of the initial $J^{\prime }$ distribution ${\mathrm{\Delta }}_{J^{\prime }}$ is held fixed; see text.) (b) Phase diagram of the ${\mathrm{hJJ}}^{\prime }$ model in the $\delta \text{−}\beta$ plane for several values of ${\mathrm{\Delta }}_{J^{\prime }}$. (c) Size of the largest cluster $s_{\max }$ as a function of $\beta$ for several values of the system size $L$. The transition becomes sharper and sharper as $L$ increases. The inset shows a finite-size scaling collapse of the same data. (Error bars indicate uncertainty due to disorder averaging; uncertainty due to Monte Carlo sampling is always smaller.)

Figure 4

(a) Low-frequency heat conductivity plotted as a function of temperature. At the phase transition, the system becomes least localized, which is associated with a cusp in the heat conductivity. Inset: Same data plotted on a log-linear scale supporting the form $\kappa \sim e^{-\text{const}|\beta -{\beta }_c|}$. (b) Heat conductivity plotted as a function of frequency for five different temperatures. The low-frequency heat conductivity displays power-law scaling with frequency $\kappa (\omega )\sim {\omega }^{\alpha }$. Inset: $\alpha$ as a function of $\beta$ (see text). [In all cases, ${\delta }_I=-0.555$ and ${\mathrm{\Delta }}_{J^{\prime }}=-0.2$. Error bars indicate uncertainty due to disorder averaging; uncertainty due to Monte Carlo sampling is always smaller. Dashed lines in (b) indicate the region being fitted for the inset.]

Figure 5

(a) Comparison of the single quasiparticle spectrum for a representative 1000 spin $hJ$ chain at criticality computed using RSRG-X and exact diagonalization (ED). (b) Comparison of the distribution of quasiparticle energies obtained using RSRG-X and ED for the hJ model. The data were obtained from 50 disorder realizations of 1000 spin chains at criticality.

Figure 6

(a)–(c) Comparison of the 1024 many-body eigenvalues computed using RSRG-X and ED for three representative 10-spin ${\mathrm{hJJ}}^{\prime }$ chains with weak, medium, and strong disorder. (d)–(f) Zoom in on spectra shown in (a)–(c). (g)–(i) Overlap of RSRG-X and ED eigenfunctions. The average overlap for the given disorder realization is indicated in the bottom of the figure (the disorder averaged overlap, frrm Fig. 7, is indicated in brackets). (See text for details.)

Figure 7

Histogram of the average overlap of RSRG-X and ED eigenfunctions, computed for three values of disorder strength. Also shown is the histogram for the null hypothesis: average overlap of ED eigenfunctions computed for pairs of different disorder realizations. (See text for details.)